Fig. 1. Schematic diagram of the designed MHMS, where the structural parameters of a unit cell (inset at the top left) are set as p=8  mm, a=6  mm, b=5  mm, w=0.5  mm; the thicknesses of the substrate and copper are 1 and 0.035 mm, respectively. Insets at the top right represent the topological transition of surface waves when the top layer rotates by an angle Δθ relative to the bottom layer at a specific frequency.

Fig. 2. Electromagnetic properties of the individual HMS. (a) Normalized transmittance and reflectance spectra when the incident waves are polarized in the x and y directions, respectively. (b), (c) Electric field distributions when the incident microwaves propagate along the x and y axes, respectively. (d), (e) Equivalent circuit models corresponding to the electric dipoles with a dipole moment along the x and y directions, respectively. (f) Calculated relative permittivity based on EMT. (g) Band diagram of the first three energy bands in the first Brillouin zone. (h), (i) The EFCs (in units of GHz) corresponding to the first two modes in the first Brillouin zone.

Fig. 3. Electromagnetic properties of the proposed bilayer HMS when Δθ=0°. (a) Normalized transmittance and reflectance spectra when the incident waves are polarized in the x and y directions, respectively. (b), (c) Calculated relative permittivity and permeability based on EMT, respectively. (d)–(g) Ez field distributions on the top surface of a unit cell at the resonance valleys corresponding to (a). (h)–(k) Current distributions on the surface of the metal between two dielectric layers at the resonance valleys of transmittance corresponding to (a). (l) Band diagrams of the first three energy bands in the first Brillouin zone. (m), (n) The EFCs (in units of GHz) corresponding to the first two modes in the first Brillouin zone.

Fig. 4. Electromagnetic properties of the proposed MHMS when Δθ=15°. (a), (b) Normalized transmittance spectra and reflectance spectra, where the first and second letters indicate the polarization directions of the incident and outgoing waves, respectively. The red line overlaps with the purple line in (b). (c), (d) Calculated relative permittivity based on EMT. (e), (f) Relative permeability extracted from S parameters.

Fig. 5. Twist-induced topological transition of surface plasmons. (a) Schematic illustration of the proposed MHMS, where the top layer is rotated counterclockwise with respect to the bottom layer. (b) At 5.90 GHz, Ez distributions at 0.05 mm above the top surface when the rotation angles Δθ are 0°, 15°, 30°, and 45°, respectively. (c) White curves represent the numerically calculated dispersion contours via Fourier transform; red and yellow dotted curves are the simulated dispersion contours of the top and bottom layers, respectively. Red arrows denote the directions of the wave vectors. (d) Simulated magic angle as a function of frequency and rotation angle. The black dotted line is the magic angle; the dispersion is hyperbolic below this line and elliptical above this line.

Fig. 6. Top view of the experimental process. A vector network analyzer was employed to generate the excitation signals; the 3D movement platform detected the Ez field distributions above the rotated HMS point by point.

Fig. 7. Experimental verifications of SPP propagation in the MHMS. (a) Detail of the measurement process, including the microwave dipole source and the scanning range (red dotted area). (b) At 6.53 GHz, measured Ez distributions in the x−y plane at 0.05 mm above the top layer when the rotation angles are 0°, 30°, 45°, and 60°, respectively. (c) Measured topological transition angles.

Fig. 8. Schematic diagram of the bilayer hyperbolic metasurface. The coordinate systems of the bottom layer and top layer are set as xyz and x′y′z, respectively, and the x′′y′′z coordinates are built to describe the wave vectors where the azimuthal angle α is defined as the one between the x and y′′ directions.

Fig. 9. Schematic diagram of the proposed moiré metasurface. Similar to Fig. 8, xyz and x′y′z are set as the coordinate systems of the bottom and top layers, respectively, while the x′′′y′′′z coordinates are used to describe the direction of the surface waves where the azimuthal angle β is defined as the one between the x and y′′′ directions.

Fig. 10. (a), (b) Simulated and measured Ez distributions when the rotation angles are 0°, 15°, 30°, and 45°, respectively. (c), (d) Calculated and measured transition processes at different frequencies and a certain rotation angle (Δθ=30°).

Fig. 11. Tunable dispersion of the bilayer structure by changing the inter-stack distance. (a) Schematic illustration of the bilayer metasurface where the top and bottom layers are separated by d. (b) Ez distributions at different gap distances d=0, 0.25, 0.5, 2.0, and 5.0 mm, and with the single top layer structure, respectively. (c) Dispersion contours via Fourier transform when the distances are 0, 0.25, 2.0, and 5.0 mm, respectively.

Fig. 12. Dispersion contours of the measured Ez distributions based on Fourier transform when (a) Δθ=0°, (b) Δθ=30°, (c) Δθ=45°, and (d) Δθ=60°.